Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Average Error: 28.9 → 9.7

Time: 58.2s

Precision: binary64

Cost: 14352

Math

\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

↓

\[\begin{array}{l} t_1 := y \cdot \left(z + y \cdot x\right)\\ t_2 := y \cdot \left(y + a\right) + b\\ t_3 := y \cdot t_2\\ t_4 := y \cdot \left(c + t_3\right) + i\\ t_5 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ t_6 := t_2 \cdot t_2\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+215}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{+40}:\\ \;\;\;\;{\left(\frac{1 + \frac{a}{y}}{x}\right)}^{-1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{t}{t_4} + \frac{y \cdot \left(230661.510616 - y \cdot \left(-27464.7644705 - t_1\right)\right)}{t_4}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+112}:\\ \;\;\;\;\frac{27464.7644705 + t_1}{t_2} + \left(230661.510616 \cdot \frac{1}{t_3} + c \cdot \left(\left(\frac{1}{t_2 \cdot t_3} \cdot -27464.7644705 + \left(230661.510616 \cdot \frac{-1}{t_2 \cdot \left(t_2 \cdot {y}^{2}\right)} - \frac{y \cdot x}{t_6}\right)\right) - \frac{z}{t_6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

↓

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* y (+ z (* y x))))
        (t_2 (+ (* y (+ y a)) b))
        (t_3 (* y t_2))
        (t_4 (+ (* y (+ c t_3)) i))
        (t_5 (+ (/ z y) (- x (/ a (/ y x)))))
        (t_6 (* t_2 t_2)))
   (if (<= y -1.2e+215)
     t_5
     (if (<= y -6.2e+40)
       (pow (/ (+ 1.0 (/ a y)) x) -1.0)
       (if (<= y 1.8e+27)
         (+
          (/ t t_4)
          (/ (* y (- 230661.510616 (* y (- -27464.7644705 t_1)))) t_4))
         (if (<= y 8e+112)
           (+
            (/ (+ 27464.7644705 t_1) t_2)
            (+
             (* 230661.510616 (/ 1.0 t_3))
             (*
              c
              (-
               (+
                (* (/ 1.0 (* t_2 t_3)) -27464.7644705)
                (-
                 (* 230661.510616 (/ -1.0 (* t_2 (* t_2 (pow y 2.0)))))
                 (/ (* y x) t_6)))
               (/ z t_6)))))
           t_5))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = y * (z + (y * x));
	double t_2 = (y * (y + a)) + b;
	double t_3 = y * t_2;
	double t_4 = (y * (c + t_3)) + i;
	double t_5 = (z / y) + (x - (a / (y / x)));
	double t_6 = t_2 * t_2;
	double tmp;
	if (y <= -1.2e+215) {
		tmp = t_5;
	} else if (y <= -6.2e+40) {
		tmp = pow(((1.0 + (a / y)) / x), -1.0);
	} else if (y <= 1.8e+27) {
		tmp = (t / t_4) + ((y * (230661.510616 - (y * (-27464.7644705 - t_1)))) / t_4);
	} else if (y <= 8e+112) {
		tmp = ((27464.7644705 + t_1) / t_2) + ((230661.510616 * (1.0 / t_3)) + (c * ((((1.0 / (t_2 * t_3)) * -27464.7644705) + ((230661.510616 * (-1.0 / (t_2 * (t_2 * pow(y, 2.0))))) - ((y * x) / t_6))) - (z / t_6))));
	} else {
		tmp = t_5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
end function

↓

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_1 = y * (z + (y * x))
    t_2 = (y * (y + a)) + b
    t_3 = y * t_2
    t_4 = (y * (c + t_3)) + i
    t_5 = (z / y) + (x - (a / (y / x)))
    t_6 = t_2 * t_2
    if (y <= (-1.2d+215)) then
        tmp = t_5
    else if (y <= (-6.2d+40)) then
        tmp = ((1.0d0 + (a / y)) / x) ** (-1.0d0)
    else if (y <= 1.8d+27) then
        tmp = (t / t_4) + ((y * (230661.510616d0 - (y * ((-27464.7644705d0) - t_1)))) / t_4)
    else if (y <= 8d+112) then
        tmp = ((27464.7644705d0 + t_1) / t_2) + ((230661.510616d0 * (1.0d0 / t_3)) + (c * ((((1.0d0 / (t_2 * t_3)) * (-27464.7644705d0)) + ((230661.510616d0 * ((-1.0d0) / (t_2 * (t_2 * (y ** 2.0d0))))) - ((y * x) / t_6))) - (z / t_6))))
    else
        tmp = t_5
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

↓

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = y * (z + (y * x));
	double t_2 = (y * (y + a)) + b;
	double t_3 = y * t_2;
	double t_4 = (y * (c + t_3)) + i;
	double t_5 = (z / y) + (x - (a / (y / x)));
	double t_6 = t_2 * t_2;
	double tmp;
	if (y <= -1.2e+215) {
		tmp = t_5;
	} else if (y <= -6.2e+40) {
		tmp = Math.pow(((1.0 + (a / y)) / x), -1.0);
	} else if (y <= 1.8e+27) {
		tmp = (t / t_4) + ((y * (230661.510616 - (y * (-27464.7644705 - t_1)))) / t_4);
	} else if (y <= 8e+112) {
		tmp = ((27464.7644705 + t_1) / t_2) + ((230661.510616 * (1.0 / t_3)) + (c * ((((1.0 / (t_2 * t_3)) * -27464.7644705) + ((230661.510616 * (-1.0 / (t_2 * (t_2 * Math.pow(y, 2.0))))) - ((y * x) / t_6))) - (z / t_6))));
	} else {
		tmp = t_5;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)

↓

def code(x, y, z, t, a, b, c, i):
	t_1 = y * (z + (y * x))
	t_2 = (y * (y + a)) + b
	t_3 = y * t_2
	t_4 = (y * (c + t_3)) + i
	t_5 = (z / y) + (x - (a / (y / x)))
	t_6 = t_2 * t_2
	tmp = 0
	if y <= -1.2e+215:
		tmp = t_5
	elif y <= -6.2e+40:
		tmp = math.pow(((1.0 + (a / y)) / x), -1.0)
	elif y <= 1.8e+27:
		tmp = (t / t_4) + ((y * (230661.510616 - (y * (-27464.7644705 - t_1)))) / t_4)
	elif y <= 8e+112:
		tmp = ((27464.7644705 + t_1) / t_2) + ((230661.510616 * (1.0 / t_3)) + (c * ((((1.0 / (t_2 * t_3)) * -27464.7644705) + ((230661.510616 * (-1.0 / (t_2 * (t_2 * math.pow(y, 2.0))))) - ((y * x) / t_6))) - (z / t_6))))
	else:
		tmp = t_5
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

↓

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(y * Float64(z + Float64(y * x)))
	t_2 = Float64(Float64(y * Float64(y + a)) + b)
	t_3 = Float64(y * t_2)
	t_4 = Float64(Float64(y * Float64(c + t_3)) + i)
	t_5 = Float64(Float64(z / y) + Float64(x - Float64(a / Float64(y / x))))
	t_6 = Float64(t_2 * t_2)
	tmp = 0.0
	if (y <= -1.2e+215)
		tmp = t_5;
	elseif (y <= -6.2e+40)
		tmp = Float64(Float64(1.0 + Float64(a / y)) / x) ^ -1.0;
	elseif (y <= 1.8e+27)
		tmp = Float64(Float64(t / t_4) + Float64(Float64(y * Float64(230661.510616 - Float64(y * Float64(-27464.7644705 - t_1)))) / t_4));
	elseif (y <= 8e+112)
		tmp = Float64(Float64(Float64(27464.7644705 + t_1) / t_2) + Float64(Float64(230661.510616 * Float64(1.0 / t_3)) + Float64(c * Float64(Float64(Float64(Float64(1.0 / Float64(t_2 * t_3)) * -27464.7644705) + Float64(Float64(230661.510616 * Float64(-1.0 / Float64(t_2 * Float64(t_2 * (y ^ 2.0))))) - Float64(Float64(y * x) / t_6))) - Float64(z / t_6)))));
	else
		tmp = t_5;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
end

↓

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = y * (z + (y * x));
	t_2 = (y * (y + a)) + b;
	t_3 = y * t_2;
	t_4 = (y * (c + t_3)) + i;
	t_5 = (z / y) + (x - (a / (y / x)));
	t_6 = t_2 * t_2;
	tmp = 0.0;
	if (y <= -1.2e+215)
		tmp = t_5;
	elseif (y <= -6.2e+40)
		tmp = ((1.0 + (a / y)) / x) ^ -1.0;
	elseif (y <= 1.8e+27)
		tmp = (t / t_4) + ((y * (230661.510616 - (y * (-27464.7644705 - t_1)))) / t_4);
	elseif (y <= 8e+112)
		tmp = ((27464.7644705 + t_1) / t_2) + ((230661.510616 * (1.0 / t_3)) + (c * ((((1.0 / (t_2 * t_3)) * -27464.7644705) + ((230661.510616 * (-1.0 / (t_2 * (t_2 * (y ^ 2.0))))) - ((y * x) / t_6))) - (z / t_6))));
	else
		tmp = t_5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]}, Block[{t$95$3 = N[(y * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y * N[(c + t$95$3), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$5 = N[(N[(z / y), $MachinePrecision] + N[(x - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 * t$95$2), $MachinePrecision]}, If[LessEqual[y, -1.2e+215], t$95$5, If[LessEqual[y, -6.2e+40], N[Power[N[(N[(1.0 + N[(a / y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[y, 1.8e+27], N[(N[(t / t$95$4), $MachinePrecision] + N[(N[(y * N[(230661.510616 - N[(y * N[(-27464.7644705 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+112], N[(N[(N[(27464.7644705 + t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(230661.510616 * N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(N[(N[(1.0 / N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision] * -27464.7644705), $MachinePrecision] + N[(N[(230661.510616 * N[(-1.0 / N[(t$95$2 * N[(t$95$2 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y * x), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.2e215 or 7.9999999999999994e112 < y

   1. Initial program 64.0

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in y around inf 13.8

      \[\leadsto \color{blue}{\left(\frac{z}{y} + x\right) - \frac{a \cdot x}{y}} \]

   3. Simplified 9.2

      \[\leadsto \color{blue}{\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)} \]
      Proof

      [Start] 13.8	\[ \left(\frac{z}{y} + x\right) - \frac{a \cdot x}{y} \]
      associate--l+ [=>] 13.8	\[ \color{blue}{\frac{z}{y} + \left(x - \frac{a \cdot x}{y}\right)} \]
      associate-/l* [=>] 9.2	\[ \frac{z}{y} + \left(x - \color{blue}{\frac{a}{\frac{y}{x}}}\right) \]

   if -1.2e215 < y < -6.1999999999999995e40

   1. Initial program 60.2

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Simplified 60.2

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}} \]
      Proof

      [Start] 60.2	\[ \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
      fma-def [=>] 60.2	\[ \frac{\color{blue}{\mathsf{fma}\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
      fma-def [=>] 60.2	\[ \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705, y, 230661.510616\right)}, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
      fma-def [=>] 60.2	\[ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x \cdot y + z, y, 27464.7644705\right)}, y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
      fma-def [=>] 60.2	\[ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, y, z\right)}, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
      fma-def [=>] 60.2	\[ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\color{blue}{\mathsf{fma}\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c, y, i\right)}} \]
      fma-def [=>] 60.2	\[ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(y + a\right) \cdot y + b, y, c\right)}, y, i\right)} \]
      fma-def [=>] 60.2	\[ \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y + a, y, b\right)}, y, c\right), y, i\right)} \]

   3. Applied egg-rr 60.2

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}\right)}^{-1}} \]

   4. Taylor expanded in y around inf 31.8

      \[\leadsto {\color{blue}{\left(\left(\frac{a}{y \cdot x} + \frac{1}{x}\right) - \frac{z}{y \cdot {x}^{2}}\right)}}^{-1} \]

   5. Simplified 31.8

      \[\leadsto {\color{blue}{\left(\frac{a}{y \cdot x} + \left(\frac{1}{x} - \frac{z}{y \cdot \left(x \cdot x\right)}\right)\right)}}^{-1} \]
      Proof

      [Start] 31.8	\[ {\left(\left(\frac{a}{y \cdot x} + \frac{1}{x}\right) - \frac{z}{y \cdot {x}^{2}}\right)}^{-1} \]
      associate--l+ [=>] 31.8	\[ {\color{blue}{\left(\frac{a}{y \cdot x} + \left(\frac{1}{x} - \frac{z}{y \cdot {x}^{2}}\right)\right)}}^{-1} \]
      unpow2 [=>] 31.8	\[ {\left(\frac{a}{y \cdot x} + \left(\frac{1}{x} - \frac{z}{y \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right)}^{-1} \]

   6. Taylor expanded in x around inf 25.8

      \[\leadsto {\color{blue}{\left(\frac{1 + \frac{a}{y}}{x}\right)}}^{-1} \]

   if -6.1999999999999995e40 < y < 1.79999999999999991e27

   1. Initial program 2.4

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in t around inf 2.4

      \[\leadsto \color{blue}{\frac{t}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i} + \frac{\left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right) \cdot y}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i}} \]

   if 1.79999999999999991e27 < y < 7.9999999999999994e112

   1. Initial program 51.4

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   2. Taylor expanded in t around 0 52.8

      \[\leadsto \color{blue}{\frac{\left(230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)\right) \cdot y}{y \cdot \left(c + y \cdot \left(\left(y + a\right) \cdot y + b\right)\right) + i}} \]

   3. Taylor expanded in i around 0 43.4

      \[\leadsto \color{blue}{\frac{230661.510616 + y \cdot \left(27464.7644705 + \left(y \cdot x + z\right) \cdot y\right)}{c + y \cdot \left(\left(y + a\right) \cdot y + b\right)}} \]

   4. Taylor expanded in c around 0 33.8

      \[\leadsto \color{blue}{\frac{27464.7644705 + \left(y \cdot x + z\right) \cdot y}{\left(y + a\right) \cdot y + b} + \left(230661.510616 \cdot \frac{1}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + -1 \cdot \left(c \cdot \left(\frac{z}{\left(b + y \cdot \left(a + y\right)\right) \cdot \left(\left(y + a\right) \cdot y + b\right)} + \left(27464.7644705 \cdot \frac{1}{\left(b + y \cdot \left(a + y\right)\right) \cdot \left(y \cdot \left(\left(y + a\right) \cdot y + b\right)\right)} + \left(230661.510616 \cdot \frac{1}{\left(b + y \cdot \left(a + y\right)\right) \cdot \left({y}^{2} \cdot \left(\left(y + a\right) \cdot y + b\right)\right)} + \frac{y \cdot x}{\left(b + y \cdot \left(a + y\right)\right) \cdot \left(\left(y + a\right) \cdot y + b\right)}\right)\right)\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+215}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{+40}:\\ \;\;\;\;{\left(\frac{1 + \frac{a}{y}}{x}\right)}^{-1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{t}{y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right) + i} + \frac{y \cdot \left(230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)\right)}{y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right) + i}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+112}:\\ \;\;\;\;\frac{27464.7644705 + y \cdot \left(z + y \cdot x\right)}{y \cdot \left(y + a\right) + b} + \left(230661.510616 \cdot \frac{1}{y \cdot \left(y \cdot \left(y + a\right) + b\right)} + c \cdot \left(\left(\frac{1}{\left(y \cdot \left(y + a\right) + b\right) \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right)\right)} \cdot -27464.7644705 + \left(230661.510616 \cdot \frac{-1}{\left(y \cdot \left(y + a\right) + b\right) \cdot \left(\left(y \cdot \left(y + a\right) + b\right) \cdot {y}^{2}\right)} - \frac{y \cdot x}{\left(y \cdot \left(y + a\right) + b\right) \cdot \left(y \cdot \left(y + a\right) + b\right)}\right)\right) - \frac{z}{\left(y \cdot \left(y + a\right) + b\right) \cdot \left(y \cdot \left(y + a\right) + b\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	7176

\[\begin{array}{l} t_1 := y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right) + i\\ t_2 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -2.06 \cdot 10^{+214}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+40}:\\ \;\;\;\;{\left(\frac{1 + \frac{a}{y}}{x}\right)}^{-1}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{t}{t_1} + \frac{y \cdot \left(230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	10.8
Cost	3400

\[\begin{array}{l} t_1 := y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right) + i\\ t_2 := \frac{a}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+40}:\\ \;\;\;\;\left(\frac{z}{y} + \left(x + \frac{27464.7644705}{y \cdot y}\right)\right) + \left(\frac{x \cdot a - z}{\frac{y \cdot y}{a}} - \left(t_2 + \frac{b}{\frac{y \cdot y}{x}}\right)\right)\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+34}:\\ \;\;\;\;\frac{t}{t_1} + \frac{y \cdot \left(230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - t_2\right)\\ \end{array} \]

Alternative 3
Error	10.8
Cost	2500

\[\begin{array}{l} t_1 := \frac{a}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+40}:\\ \;\;\;\;\left(\frac{z}{y} + \left(x + \frac{27464.7644705}{y \cdot y}\right)\right) + \left(\frac{x \cdot a - z}{\frac{y \cdot y}{a}} - \left(t_1 + \frac{b}{\frac{y \cdot y}{x}}\right)\right)\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+34}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)\right)}{y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right) + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - t_1\right)\\ \end{array} \]

Alternative 4
Error	10.8
Cost	2376

\[\begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+41}:\\ \;\;\;\;\frac{z}{y} + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) - \left(\frac{x \cdot a}{y} + \left(z - x \cdot a\right) \cdot \frac{a}{y \cdot y}\right)\right)\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)\right)}{y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right) + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \end{array} \]

Alternative 5
Error	14.7
Cost	2248

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{z}{y} + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) - \left(\frac{x \cdot a}{y} + \left(z - x \cdot a\right) \cdot \frac{a}{y \cdot y}\right)\right)\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{+34}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + \left(y + a\right) \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \end{array} \]

Alternative 6
Error	16.6
Cost	2136

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{230661.510616 + z \cdot \left(y \cdot y\right)}{c + \left(\left(y + a\right) \cdot \left(y \cdot y\right) + y \cdot b\right)}\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)}{c + y \cdot \left(y \cdot a\right)}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+33}:\\ \;\;\;\;\frac{t - y \cdot -230661.510616}{y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right) + i}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	13.0
Cost	2124

\[\begin{array}{l} t_1 := c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\\ t_2 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -3 \cdot 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)}{t_1}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+33}:\\ \;\;\;\;\frac{t - y \cdot \left(-230661.510616 - z \cdot \left(y \cdot y\right)\right)}{y \cdot t_1 + i}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	19.0
Cost	2008

\[\begin{array}{l} t_1 := c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\\ t_2 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{+41}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{230661.510616 + z \cdot \left(y \cdot y\right)}{t_1}\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-36}:\\ \;\;\;\;\frac{t}{y \cdot t_1 + i}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+33}:\\ \;\;\;\;\frac{t - y \cdot -230661.510616}{i + y \cdot \left(c + \left(y + a\right) \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	19.0
Cost	2008

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq -8.4 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{230661.510616 + z \cdot \left(y \cdot y\right)}{c + y \cdot \left(y \cdot \left(y + a\right) + b\right)}\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)}{c + y \cdot \left(y \cdot a\right)}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+33}:\\ \;\;\;\;\frac{t - y \cdot -230661.510616}{i + y \cdot \left(c + \left(y + a\right) \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	19.0
Cost	2008

\[\begin{array}{l} t_1 := \left(y + a\right) \cdot \left(y \cdot y\right)\\ t_2 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{230661.510616 + z \cdot \left(y \cdot y\right)}{c + \left(t_1 + y \cdot b\right)}\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)}{c + y \cdot \left(y \cdot a\right)}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+32}:\\ \;\;\;\;\frac{t - y \cdot -230661.510616}{i + y \cdot \left(c + t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	15.8
Cost	1996

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)}{c + y \cdot \left(y \cdot \left(y + a\right) + b\right)}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+33}:\\ \;\;\;\;\frac{t - y \cdot \left(-230661.510616 - z \cdot \left(y \cdot y\right)\right)}{i + y \cdot \left(c + \left(y + a\right) \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	13.4
Cost	1992

\[\begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{z}{y} + \left(\left(x + \frac{27464.7644705}{y \cdot y}\right) - \left(\frac{x \cdot a}{y} + \left(z - x \cdot a\right) \cdot \frac{a}{y \cdot y}\right)\right)\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+33}:\\ \;\;\;\;\frac{t - y \cdot \left(-230661.510616 - z \cdot \left(y \cdot y\right)\right)}{y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right) + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \end{array} \]

Alternative 13
Error	21.2
Cost	1880

\[\begin{array}{l} t_1 := c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\\ t_2 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+41}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{230661.510616 + z \cdot \left(y \cdot y\right)}{t_1}\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\frac{t}{y \cdot t_1 + i}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	14.9
Cost	1864

\[\begin{array}{l} t_1 := c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\\ t_2 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{230661.510616 - y \cdot \left(-27464.7644705 - y \cdot \left(z + y \cdot x\right)\right)}{t_1}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{t - y \cdot -230661.510616}{y \cdot t_1 + i}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	21.6
Cost	1484

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{t}{y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right) + b\right)\right) + i}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	23.3
Cost	1356

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+33}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	22.3
Cost	1356

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+32}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	29.1
Cost	1100

\[\begin{array}{l} t_1 := \frac{z}{y} + \left(x - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{+123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 55000000000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	33.1
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+124}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 1600000000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 20
Error	33.2
Cost	588

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+124}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{\frac{a}{x}}\\ \mathbf{elif}\;y \leq 1850000000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 21
Error	46.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.31:\\ \;\;\;\;\frac{230661.510616}{c}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 22
Error	32.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 23
Error	47.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023023 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))